Summary
This note provides an elementary short proof of the Knaster-Kuratowski-Mazurkiewicz-Shapley (K-K-M-S) Theorem based on Brouwer's fixed point theorem. The usefulness of the K-K-M-S Theorem lies in the fact that it can be applied to prove directly Scarf's (1967) Theorem, i.e. any balanced game has a non-empty core. We also show that the K-K-M-S Theorem and the Gale-Nikaido-Debreu Theorem can be proved by the same arguments.
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We wish to thank Roko Aliprantis for useful comments.
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Krasa, S., Yannelis, N.C. An elementary proof of the Knaster-Kuratowski-Mazurkiewicz-Shapley Theorem. Econ Theory 4, 467–471 (1994). https://doi.org/10.1007/BF01215384
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DOI: https://doi.org/10.1007/BF01215384